Explicit latent variable models provide a flexible yet powerful framework for data synthesis, enabling controlled manipulation of generative factors. With latent variables drawn from a tractable probability density function that can be further constrained, these models enable continuous and semantically rich exploration of the output space by navigating their latent spaces. Structured latent representations are typically obtained through the joint minimization of regularization loss functions. In variational information bottleneck models, reconstruction loss and Kullback-Leibler Divergence (KLD) are often linearly combined with an auxiliary Attribute-Regularization (AR) loss. However, balancing KLD and AR turns out to be a very delicate matter. When KLD dominates over AR, generative models tend to lack controllability; when AR dominates over KLD, the stochastic encoder is encouraged to violate the standard normal prior. We explore this trade-off in the context of symbolic music generation with explicit control over continuous musical attributes. We show that existing approaches struggle to jointly minimize both regularization objectives, whereas suitable attribute transformations can help achieve both controllability and regularization of the target latent dimensions.

Bayesian inference, while foundational to probabilistic reasoning, is often hampered by the computational intractability of posterior distributions, particularly through the challenging evidence integral. Conventional approaches like Markov Chain Monte Carlo (MCMC) and Variational Inference (VI) face significant scalability and efficiency limitations. This paper introduces a novel, unifying framework for fast Bayesian updates by leveraging harmonic analysis. We demonstrate that representing the prior and likelihood in a suitable orthogonal basis transforms the Bayesian update rule into a spectral convolution. Specifically, the Fourier coefficients of the posterior are shown to be the normalized convolution of the prior and likelihood coefficients. To achieve computational feasibility, we introduce a spectral truncation scheme, which, for smooth functions, yields an exceptionally accurate finite-dimensional approximation and reduces the update to a circular convolution. This formulation allows us to exploit the Fast Fourier Transform (FFT), resulting in a deterministic algorithm with O(N log N) complexity -- a substantial improvement over the O(N^2) cost of naive methods. We establish rigorous mathematical criteria for the applicability of our method, linking its efficiency to the smoothness and spectral decay of the involved distributions. The presented work offers a paradigm shift, connecting Bayesian computation to signal processing and opening avenues for real-time, sequential inference in a wide class of problems.

Large Language Models (LLMs) have demonstrated remarkable capabilities across numerous tasks, yet principled explanations for their underlying mechanisms and several phenomena, such as scaling laws, hallucinations, and related behaviors, remain elusive. In this work, we revisit the classical relationship between compression and prediction, grounded in Kolmogorov complexity and Shannon information theory, to provide deeper insights into LLM behaviors. By leveraging the Kolmogorov Structure Function and interpreting LLM compression as a two-part coding process, we offer a detailed view of how LLMs acquire and store information across increasing model and data scales -- from pervasive syntactic patterns to progressively rarer knowledge elements. Motivated by this theoretical perspective and natural assumptions inspired by Heap's and Zipf's laws, we introduce a simplified yet representative hierarchical data-generation framework called the Syntax-Knowledge model. Under the Bayesian setting, we show that prediction and compression within this model naturally lead to diverse learning and scaling behaviors observed in LLMs. In particular, our theoretical analysis offers intuitive and principled explanations for both data and model scaling laws, the dynamics of knowledge acquisition during training and fine-tuning, factual knowledge hallucinations in LLMs. The experimental results validate our theoretical predictions.

Ordinal categorical data are routinely encountered in a wide range of practical applications. When the primary goal is to construct a regression model for ordinal outcomes, cumulative link models represent one of the most popular choices to link the cumulative probabilities of the response with a set of covariates through a parsimonious linear predictor, shared across response categories. When the number of observations grows, standard sampling algorithms for Bayesian inference scale poorly, making posterior computation increasingly challenging in large datasets. In this article, we propose three scalable algorithms for approximating the posterior distribution of the regression coefficients in cumulative probit models relying on Variational Bayes and Expectation Propagation. We compare the proposed approaches with inference based on Markov Chain Monte Carlo, demonstrating superior computational performance and remarkable accuracy; finally, we illustrate the utility of the proposed algorithms on a challenging case study to investigate the structure of a criminal network.

The quantity of interest in the classical Cramér-Rao theory of unbiased estimation (e.g., the Cramér-Rao lower bound, its exact attainment for exponential families, and asymptotic efficiency of maximum likelihood estimation) is the variance, which represents the instability of an estimator when its value is compared to the value for an independently-sampled data set from the same distribution. In this paper we are interested in a quantity which represents the instability of an estimator when its value is compared to the value for an infinitesimal additive perturbation of the original data set; we refer to this as the "sensitivity" of an estimator. The resulting theory of sensitivity is based on the Wasserstein geometry in the same way that the classical theory of variance is based on the Fisher-Rao (equivalently, Hellinger) geometry, and this insight allows us to determine a collection of results which are analogous to the classical case: a Wasserstein-Cramér-Rao lower bound for the sensitivity of any unbiased estimator, a characterization of models in which there exist unbiased estimators achieving the lower bound exactly, and some concrete results that show that the Wasserstein projection estimator achieves the lower bound asymptotically. We use these results to treat many statistical examples, sometimes revealing new optimality properties for existing estimators and other times revealing entirely new estimators.

We establish parameter inference for the Poisson canonical polyadic (PCP) tensor model through a latent-variable formulation. Our approach exploits the observation that any random PCP tensor can be derived by marginalizing an unobservable random tensor of one dimension larger. The loglikelihood of this larger dimensional tensor, referred to as the "complete" loglikelihood, is comprised of multiple rank one PCP loglikelihoods. Using this methodology, we first derive non-iterative maximum likelihood estimators for the PCP model and demonstrate that several existing algorithms for fitting non-negative matrix and tensor factorizations are Expectation-Maximization algorithms. Next, we derive the observed and expected Fisher information matrices for the PCP model. The Fisher information provides us crucial insights into the well-posedness of the tensor model, such as the role that tensor rank plays in identifiability and indeterminacy. For the special case of rank one PCP models, we demonstrate that these results are greatly simplified.

In high-stakes applications, predictive models must not only produce accurate predictions but also quantify and communicate their uncertainty. Reject-option prediction addresses this by allowing the model to abstain when prediction uncertainty is high. Traditional reject-option approaches focus solely on aleatoric uncertainty, an assumption valid only when large training data makes the epistemic uncertainty negligible. However, in many practical scenarios, limited data makes this assumption unrealistic. This paper introduces the epistemic reject-option predictor, which abstains in regions of high epistemic uncertainty caused by insufficient data. Building on Bayesian learning, we redefine the optimal predictor as the one that minimizes expected regret -- the performance gap between the learned model and the Bayes-optimal predictor with full knowledge of the data distribution. The model abstains when the regret for a given input exceeds a specified rejection cost. To our knowledge, this is the first principled framework that enables learning predictors capable of identifying inputs for which the training data is insufficient to make reliable decisions.

Embedding non-restrictive prior knowledge, such as energy conservation laws, in learning-based approaches is a key motive to construct physically consistent models from limited data, relevant for, e.g., model-based control. Recent work incorporates Hamiltonian dynamics into Gaussian Process (GP) regression to obtain uncertainty-quantifying models that adhere to the underlying physical principles. However, these works rely on velocity or momentum data, which is rarely available in practice. In this paper, we consider dynamics learning with non-conservative Hamiltonian GPs, and address the more realistic problem setting of learning from input-output data. We provide a fully Bayesian scheme for estimating probability densities of unknown hidden states, of GP hyperparameters, as well as of structural hyperparameters, such as damping coefficients. Considering the computational complexity of GPs, we take advantage of a reduced-rank GP approximation and leverage its properties for computationally efficient prediction and training. The proposed method is evaluated in a nonlinear simulation case study and compared to a state-of-the-art approach that relies on momentum measurements.

The loss function used to train a neural network is strongly connected to its output layer from a statistical point of view. This technical report analyzes common activation functions for a neural network output layer, like linear, sigmoid, ReLU, and softmax, detailing their mathematical properties and their appropriate use cases. A strong statistical justification exists for the selection of the suitable loss function for training a deep learning model. This report connects common loss functions such as Mean Squared Error (MSE), Mean Absolute Error (MAE), and various Cross-Entropy losses to the statistical principle of Maximum Likelihood Estimation (MLE). Choosing a specific loss function is equivalent to assuming a specific probability distribution for the model output, highlighting the link between these functions and the Generalized Linear Models (GLMs) that underlie network output layers. Additional scenarios of practical interest are also considered, such as alternative output encodings, constrained outputs, and distributions with heavy tails.

Pedestrian inertial localization is key for mobile and IoT services because it provides infrastructure-free positioning. Yet most learning-based methods depend on fixed sliding-window integration, struggle to adapt to diverse motion scales and cadences, and yield inconsistent uncertainty, limiting real-world use. We present ReNiL, a Bayesian deep-learning framework for accurate, efficient, and uncertainty-aware pedestrian localization. ReNiL introduces Inertial Positioning Demand Points (IPDPs) to estimate motion at contextually meaningful waypoints instead of dense tracking, and supports inference on IMU sequences at any scale so cadence can match application needs. It couples a motion-aware orientation filter with an Any-Scale Laplace Estimator (ASLE), a dual-task network that blends patch-based self-supervision with Bayesian regression. By modeling displacements with a Laplace distribution, ReNiL provides homogeneous Euclidean uncertainty that integrates cleanly with other sensors. A Bayesian inference chain links successive IPDPs into consistent trajectories. On RoNIN-ds and a new WUDataset covering indoor and outdoor motion from 28 participants, ReNiL achieves state-of-the-art displacement accuracy and uncertainty consistency, outperforming TLIO, CTIN, iMoT, and RoNIN variants while reducing computation. Application studies further show robustness and practicality for mobile and IoT localization, making ReNiL a scalable, uncertainty-aware foundation for next-generation positioning.